No, you cannot use non-integer counts with edgeR.
If you must use non-integer counts, please use the voom() function in the limma package instead. This will do an analysis that is not too different from edgeR, just a little less powerful, and is not bothered by non-integer values.
Edit: more recent versions of edgeR do now allow non-integer counts. However the values must be still be on the same scale as the counts, e.g., the values should add up approximately to the correct library sizes. So it is still not suitable for your weighted counts.
> Date: Mon, 21 May 2012 12:24:43 -0700
> From: Mete Civelek <firstname.lastname@example.org>
> To: <email@example.com>
> Subject: [BioC] non-integer counts for edgeR
> Dear All,
> I am analyzing microRNAseq data with edgeR. Because some of the reads map
> to multiple locations, I weighed the counts based on the number of genomic
> locations that a read maps. For example, if a read maps to 3 locations,
> each location gets a count of 1/3. Of course, this means that the read counts
> for some miRNAs in some samples are not integers. Is it possible to use these
> counts to do differential expression analysis with a quantitative trait in
> edgeR? My understanding is "no" but I thought someone can suggest a
> Thank you for your help.
> Best Regards,